Network Traffic Randomness for Cryptographic Purposes
Nathaniel Puffer
James Madison University
CS 627
Dr. Charles Abzug
April 30, 2002
Abstract
Random numbers surround us in our daily activities, in gaming, chance occurrences, or in
formulaic propagation. Within cryptography random numbers serve as the seed for
formulas creating non-predictable sequences necessary for security. The problem is in the
generation of these numbers.
Though randomness is well understood and standard metrics exist for its measure,
creating randomness in an artificial environment with the appropriate quantity of output
can be problematic. Computers have the ability to produce strings of numbers based on
complex conditions which create numbers that appear random. However, due to their
programmed nature these numbers are only pseudo-random, and can be replicated or
predicted by other machines. Rolling of dice or selection of cards are more truly random
events, but must be done by hand and do not have sufficient yield. Radiation emission is
known to be truly random, but poses a logistical problem in its containment and
detection.
Additional natural random number generators have been devised based on radio noise
light emissions. The purpose of this paper is to explore the packet traffic readily available
to any machine connected to the internet. Within that frame of reference, it is proposed
that an ICMP ping packet traveling a significant distance will encounter sufficient
random fluctuations in routing that its return time will be unpredictable. To this end the
availability of this random source will allow a sufficient seed to easily create one time
keypads, the most secure cryptographic mechanism.
Introduction
The internet has grown from 15 nodes in 1970s to millions of nodes today. With each
exponential increase a matching increase in complexity occurs in the infrastructure
involved to interconnect the host systems. At each juncture within this infrastructure
thousands of computations occur dictating the quality of service within a datagram’s
frame of reference. The primary protocol for delivery, TCP/IP, is a “best effort” service,
limiting guarantees and increasing the magnitude of random occurrences.
Finally, the state of the internet is not constant. As each host makes a different request the
overall status of in transit packets changes hundreds of times a second. Therefore the
round trip time for any ping request should pass through thousands of non repeatable
states on its journey. The combination of all of these factors should create a highly
complex environment yielding unpredictable results.
This environment is deemed necessary for the output of truly random numbers for the
sake of security. RFC 1750, Randomness Recommendations for Security, indicates that a
few hundred bits of seed material that is both random and truly unpredictable can be
sufficient to produce cryptographically strong sequences. However, for the sake of a One
Time Pad it would be desirable to harvest as large a seed as possible.
Materials and Methods
In order to test the randomness of the ping packets three network samples were taken.
The first sample was from the localhost [134.126.69.198 at JMU], which was to serve as
a baseline of a highly predictable network occurrence. The second sample was taken from
ping packets to http://www.uva.edu. This site serves as a non local host but does not
transcend top level routers. The third site was http://www.ucla.edu. This site was chosen
for its distance from the originating host without crossing international router boundaries.
Each sample was run for 1000 iterations to form a large enough test population for
statistical analysis. Two additional baseline measurements were made, one using the
rand() function within excel and modified to the appropriate range, and another using
atmospheric radio emissions.
In order to create relativistic boundaries for the experiment the lower baseline should be
the locahost data, providing little if any randomness. The high level baseline should be
the atmospheric data, providing a high level of randomness. The RAND() function should
fall between these two values with its pseudorandom nature.
Each one of the test runs was executed on an OpenBSD system and piped directly to a
flat text file, shown in Example 1.
icmp_seq=46 ttl=239 time=85.260 ms
64 bytes from 169.232.33.130: icmp_seq=54 ttl=239 time=88.998 ms
64 bytes from 169.232.33.130: icmp_seq=55 ttl=239 time=81.935 ms
64 bytes from 169.232.33.130: icmp_seq=56 ttl=239 time=82.076 ms
64 bytes from 169.232.33.130: icmp_seq=57 ttl=239 time=90.000 ms
Example 1. Raw Data
Each of the flat files was then imported into Excel. During this import only the 1/1000 of
the ms were retained, shown in Example 2. This was done to preserve the most random
part of the sequence and eliminate repeating characters.
Example 2. Stripped Data
Each of the files was then exported back to flat text, removing all spaces and carriage
returns to prep them from statistical analysis. A sample of this data block is shown in
Example 3.
92981853438873251392103622804762327579065952763872556655454478779613905
63845847120573526617297780522622502987448418848515475275229914395818836
63925792344854458658296265114860990111931205545860768320916981222414152
05075304171395751384791158849302632
Example 3. Data Block
ENT, a pseudorandom number sequence test program was configured to interpret each
test set. This program provided for the execution of a variety of tests on the data. Each
file was interpreted as a stream of bytes and produced an output file with the signature
show in Example 4.
Entropy = 7.980627 bits per character.
Optimum compression would reduce the size of this 51768 character file by 0 percent.
Chi square distribution for 51768 samples is 1542.26, and randomly would exceed this
value 0.01 percent of the times.
Arithmetic mean value of data bytes is 125.93 (127.5 = random).
Monte Carlo value for Pi is 3.169834647 (error 0.90 percent).
Serial correlation coefficient is 0.004249 (totally uncorrelated = 0.0).
Example 4. ENT Sample Output
The entropy measurement is the information density of the contents of the file, expressed
as a number of bits per character. Highly entropic files are relatively random. In addition
the compression metric is given to indicate the quantity of non-entropic space within the
file.
The second measurement is the chi-square test, and is the most commonly used test for
the randomness of data due to its extreme sensitivity to errors in pseudorandom number
sequences.
The chi-square distribution is calculated by taking the stream of bytes in the file and
expressing them as an absolute number. In addition a percentage is calculated showing
how frequently a truly random sequence would exceed the value calculated.
Interpretation of the data is done using the percentage ranges shown in Table 1.
n < 1%
5% < n < 1%
10% < n < 5%
90% < n < 10%
95% < n < 90%
99% < n < 95%
n > 99%
not random
suspect
almost suspect
random
almost suspect
suspect
not random
Table 1. Chi Test Interpretations
As a reference for the stringency of this test the standard Unix rand() function returns a
Chi square distribution of .01, and is unacceptable non random.
The arithmetic mean is the result of summing the all the bytes in the data file divided by
the file length. If the data are close to random, this should be approximately .5. If the
mean departs from this value, the values are consistently high or low, allowing division
of the original set for further statistical analysis.
ENT also calculates a Monte Carlo value of Pi for each successive sequence of six bytes.
These six bytes are used as 24 bit X and Y co-ordinates within a square. If the distance of
the randomly-generated point is less than the radius of a circle inscribed within the
square, the six-byte sequence is considered a "hit". Using a percentage of hits to calculate
Pi a very random data set should approach Pi. This test in contingent upon large data sets.
The final calculation is the serial correlation coefficient. This quantity measures the
extent to which each byte in the file depends upon the previous byte. For highly random
sequences this values should approach zero, with no serialization within the byte stream.
Highly predictable sequences should have values approaching 1.
Results and Interpretations
After running each data set through the ENT analyzer all of the results were tabulated and
summarized into Table 2.
Data Set
localhost
UVA
UCLA
RAND()
Random.org
Entropy
0.990924
0.991358
0.993697
0.990958
0.990924
Compression
0.00%
0.00%
0.00%
0.00%
0.00%
Chi
Square
0.01
0.01
0.01
0.01
0.01
Arithmetic
Mean
0.444
0.4453
0.4533
0.4441
0.444
Monte
Carlo
4
4
4
4
4
SCC
0.016484
0.014082
0.007695
0.028486
0.016484
Table 2. Data Summary
Each of the data sets showed a low level of entropy which can be interpreted as a low
level of variance in the bit patterns of the file. Since the file was originally numerical data
from 0 to 9 the bit signatures should not vary dramatically. Since all spaces were
removed from the files the lack of additional compression is expected. The Chi Square
values are all unacceptably low, indicating that the file is not random. Monte Carlo Pi
approximation was also very poor.
However, both the arithmetic mean and the SCC were in acceptable ranges. This means
that the numbers were well distributed and there was a low level of predictability between
them.
In addition a graphical analysis of the data sets was done in an attempt to visually
determine if patterns existed within the files. The results of this analysis are shown in
Graph 1 through Graph 5.
Graph 1. Localhost Data
localhost
350
300
1/1000 ms
250
200
150
100
50
0
1
42
83
124 165 206 247 288 329 370 411 452 493 534 575 616 657 698 739 780 821 862 903 944 985
packet
The localhost data is as expected with little variation throughout the timeline. This
relatively flat event shows a highly predictable pattern of numbers within the available
range.
Graph 2. UVa Data
UVa
1200
1000
1/1000 ms
800
600
400
200
0
1
42
83
124 165 206 247 288 329 370 411 452 493 534 575 616 657 698 739 780 821 862 903 944 985
ping count
The UVa data shows a much higher magnitude of randomness throughout the cycle.
However there is a banding effect around .600 ms that shows a disproportionate
frequency. Thought the ENT tests did not show a drastically different result from the
localhost data, this banding raises concerns about the predictability of the set.
Graph 3. UCLA Data
UCLA
1200
1000
1/1000 ms
800
600
400
200
0
1
42
83
124 165 206 247 288 329 370 411 452 493 534 575 616 657 698 739 780 821 862 903 944 985
packet number
Like the UVa data set, the UCLA data also shows banding around the .775ms and .925ms
ranges. At this time the specific cause of the banding is not known, but is distinct enough
to warrant future study.
Graph 4. Rand() Data
Rand()
1200
1000
value
800
600
400
200
0
1
42
83
124 165 206 247 288 329 370 411 452 493 534 575 616 657 698 739 780 821 862 903 944 985
count
Though it is well known that the RAND() function will only yield pseudorandom
numbers we know that the distribution is not in fact random. However, the graphical
distribution appears to be random.
Graph 5. Random.org Data
Random.org
1200
1000
value
800
600
400
200
0
1
42
83
124 165 206 247 288 329 370 411 452 493 534 575 616 657 698 739 780 821 862 903 944 985
count
The data from Random.org is known to be truly random, and creates that impression
through it’s graphical representation.
Discussion
Within the original sets of data certain characteristics were known. The localhost data
should be almost perfectly predictable, while the data obtained from Random.org should
have been almost perfectly random. The UVa., UCLA, and RAND() data should have
been in between the two baseline benchmarks.
Though these conclusions were shown with the graphical representation of the data the
statistical analysis did not bear out the same conclusions. The results of the ENT showed
that all of the data sets were basically the same as far as randomness. The possible
reasons for this could be that the data sets were too small. In addition it’s feasible that the
bit format of the data was not in the appropriate form to properly analyze the number
patterns.
In the future it would be beneficial to take a larger data set on the order of 10,000
packets. Additionally these data sets should be converted to binary before they are fed
into the ENT in order to ascertain if the input format has skewed the results.
Finally, the presence of banding within the UVa. and UCLA data sets is particularly
interesting. It is theoretical that this banding represents a remnant or indication of the
number of major routing centers that the packet must traverse in order to achieve a round
trip.
A future experiment is being planned that will try to find repeatable bands with
appropriate correlations to physical or logical topography within the network.
Conclusions
Running the ENT analysis on the data sets was inconclusive in determining the quality of
randomness of network data for cryptographic means. However, predictable remnants
within the network traffic have lead to the probability that ping packets are not
sufficiently random for the purposes of key generation. Since one time key pads are
especially vulnerable to patterns within large blocks of text the lack of a positive
conclusion forces the determination that ping data cannot be used.
Bibliography
Haahr, Mads. Random.org. http://www.random.org/essay.html. Last Visited, April 20,
2002.
Schiller, Crocker, Eastlake. RFC 1750, Randomness Recommendations for Security.
http://www.cis.ohio-state.edu/cgi-bin/rfc/rfc1750.html. Last Visited, April 20, 2002.
Walker, John. Pseudorandom Number Sequence Test Program.
http://www.fourmilab.ch/random/. Last Visited, April 20, 2002.
Appendices
Appendix A. Data Blocks
Localhost
92981853438873251392103622804762327579065952763872556655454478779613905638458
47120573526617297780522622502987448418848515475275229914395818836639257923448
54458658296265114860990111931205545860768320916981222414152050753041713957513
84791158849302632213511941917908778485517397966428305149486316865411949310017
14725686459694543528198308779477881119312671597247041856717239575526128562521
42306959907175739480179486136160408318631376984174681036322737259827183059124
47111387060569725795658981643207994506310186878232793582550795769459985585574
34998593851488378939688797716048745232092157297192030259821255163696442031588
11952930332261925975969286424513279799534819091189426792029932879427716407688
08515453612721849343715817547474165568628041762793574240790552573181869648264
72708985259874219470228965169587662745409314551765081628862190340069510794886
37234517968262388971882869788596306972807590269922835316361503225870292727244
13496839512618368855133286500710147258243224691981317019457454067516215669845
56311158608114272913796667753206376640212239238415974249552091332678813553177
55019483316289571535489112638772573867889929869831544693367252167464029332437
11795604617424003507246218275367545112520957823439506242689836895158008827413
96515195259344168565879832611369159415333413747175956792622346604202641118167
45354664706762693206833951119029765377054244344466954072886221525535221999167
65945222629146187354975759954522953169249954925375621827686171044595543961635
86650649900464223818915157801192224034945265333813933170181667124469711788691
43843536221766390890895774913894684899365245118283867129847695813437329193896
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53667468285157094601034055476938064338427017463218275858371130964975730615587
89727031502984370672900824875550890838871672188425826750304624565959293166329
99973675096292769317879732327546993275606609462578753471428180891842052166302
50156550457670661249987611131311969965207275887796113285013302227126957580948
81154556874174589661162371655172188829190583834447038837695280788960605260723
54940470450249634851679275779804774151483120253329457143344051395423376118258
86631828964902271121587814528835595296114083815447885325079702309574684060537
44543304564155027215871717964667072481372697192412171644807199282374555063461
94522523055361043799568058888999936325514590576819914552538019065875124395599
39858898488604165059801962372876087684179176581175111080677774223394288256696
33972117739684918931726215820308647981389945692658963838178793903306525237274
80311798888845182333419582246539722019644355745431913329638510642275443693642
42962967248101838832094329791750864459846717499838770693930589478911150922055
67457892552987888142727523071591397124930585310551941000480372720921303591957
20469674822638605367863172165526446232311016087391365772731281681974335241870
545145561852556800925502315329746237262136434
UVa.
69797625368682559845243553736320038921262252437264885441509619319896998706187
57621749842671396753214861337261624562827769329093199399473662145725387397033
89571824637554262543943707658408719006207785221718248611962120647110199066731
57136988229342025361463051796262090091850661928292192361713253908176457946718
95267196034853199786478098546548968499936765522156861929806618531670169061921
78560399052562139560718088561967867723580886573915565587623233403438654730228
45624346173706895185346213711283386521361859759015335764147186617675650921407
48335376148168202125632512447619137926659437535713129600440616944763786617295
89862635664361772692990663038939251992751379334640264867972965662846624523978
52848136883985815184621324305618620327139919517993619618863225199278187526457
04336241652195175866205593948333156391719736877944955552762085814125013180593
24887243287583487653537723996195179576238428686191671140484319156929279848597
96828993312912323193767028511962097743942753876159731452462117388618635493618
21431350620277979436243689589306538446041291970624323354115491042770047364797
48481252620329078337662133812432462317210914542828661511849187411734414783606
05456614606999215467901507562398233617577635422272874595259471618227351622943
99442848134364383533513344937784196661392709588163156973836372954762062226196
26995247036227022722664967934224259559849668012136492975210789438029561742821
63813936221594721809826726152011953638966193578076206273033481599355319363222
86523161944705174825149486179862111056618162767771195323371334390862145438797
96207458292587899195162574856246401295287071999762392928921358162602624631634
44341962466592698852525309125946203879711328854425998706298286235911669521116
20786397503179275476227585799956253077963942345708655515155435727109245156236
18187530179618175553623457492620622997158559585094412918274403126202002096238
22708617619275996623533116442853538417862756361156154189969127562293341462133
38112379532402380945378628197582625685890650571871476415698815516686268106426
22973513370986831118352299662262712142777622368509706620183939142562494612272
19057859250636570141354414268870785689394525921716215896214105493732561863238
39022958685176209959963535145552898250410461781899887463062626961762174408318
04479969841776640274622765403524620249872221436441979619234441478336385383914
45029550531714625625548295337927432868762418772775386574261895346998266155865
50876832401618109326762455761264763336049852462553804617636592430385400244915
29428611615576576538119297435537009142684032049035436142777862190225593368578
66174981957834487147306446777007876236166014327462128461690519740479362233936
69566374361911750117740227312012062277850353775499838262496162498624699174984
23887074155688742962096482700618292474868151438162618361203165623858195551626
92014820884645625311374799958348507209802632283469537922846197039806244381463
475887846361868036199996642144557624236883717549
UCLA
64092596998293349947311576464397927791495593539770163378399435276916953650238
19184249352367663093694823692771557776694147634269527595326040154918524141827
24729989357609359983068993199494949368682039389649916353313814092754411785893
67047769368023852278657573611477169509349357593355378944031193302689349492978
66486375555169359419339362795229996655236157236729847784340599193496999973679
39676601171973692786759407860089928225732585947326939472123686497255394412609
34501223371569937887779368767159342073351293593639770793379577444393967554715
59893084324085201396339141899587887784761159337957939398299538726549610934932
43385416741712289336517519379415837519357073393513994935948285716934617117609
82935819493979780330980159456195926398649929364219942185453315547650247582134
07356341648455839354322361447517007449999317537515139741941935935629353832078
25936934392842801675321453693084088369346088649596223514869442583024288867518
49995824309361821481318886663617949320496161800251933436167214938939434273915
94710397460222488138236827506359499127957111689466568352715093410534513944192
48493450678299018671211710992832472842152635631471179789793270662111883701475
93432952594750272094177684538639123828059342412213814899367223497939590162595
65522088018754216759616976514935856943941954786617755967659293661785262956199
26356435389549319096639369549551684938707294375831136511700962213815419856938
86753702235216749749744750936858437635850946127025361569334015014367551378393
59999489999339338324721861511447504201469357045537280595862443409361805831242
74391935932642838960953423530722659766723573713612802999984937581393787626411
49372837595493696783593478678144971846090210693561267631499914054293931795979
01147547996779429801151799353309739566989205700387329292234759168249347275993
59303959055897547960893637171438219252787051793427417027676781114152955925240
94030393932265699028847631584114237995928379579197524320869802197293770126833
64012440259649669745912678035473896825146901919720251307505683421297514724461
16097251741117503921424984174393815999758821568713235859750751496963367297522
06618593226904756947936755736956741628746397419016921749848101262304543872694
57177525329111809101941751602751560233781713053513946275093449731974932687648
72092042158675169867446544599235984318273156160652750932739751756219580193463
87499971849890506149751470875706039941016170782256369756696983953753496056676
23620752460339488935935173223746860752180754672279491952631812752947668396700
99980076489356447605294392752691750518749234275175066888975093542613587781751
77847593598275063546051775074349866193559034283183388738794435860246424352404
05302165884685477539359393949399884029018835703309843715415716601923168381934
66993587094824320935737252499145710993567135164580949386357204626695879993028
041217453943814511881935439751542757533
Rand()
39986229584815498152545579053035777001367684483418035495757031009189327975776
11953394008871575395103617237279973662237208270572750045173316823644649373251
33923725186675715201312621963006059908137801469818865950290247768545277395849
08177198934548098192206748179838503868621537874413477982782447251959236936553
83882928194937736348629271159983032187960138411392233883328633980092639967170
27722596849443111451584221321585482749811763582898798541514947509520721432822
29528013866341024042125627731454351832084997269265782770330189065215352553439
76396258650423833138425297379239291375578469694591831107130465987577901515722
24181083798824489734605069631473475513403231363844688461906583950525768213395
32009911828806059647507436112359167644372107307853438137338315957422375555807
52236696556501479569215838817272082124133269799312694161708308486033978609345
71231327069499979583573753187684514798259116747866098530671885498822771876372
95071854893813671545329932876861714344364611344739660532860128449628446544749
84949552244156175822313928693593443841294233779813364827669982242897802292017
89701612681570237885983465221327324068159213608495575227525306247898977452498
70614682352253689159631407968652654224319882672117398885702812276179803506619
77314626361602429904982582658184335189885437399260195868225919064954874914465
64268095852182001476622745305016937971236293255096140687141566188847990869581
20756493392526741822512105783988509293554218858537348804968564085234224760860
16665452832755816279189997267356052940517878624529488462863546916412913181427
66449558180944218631172344069156344958751241192746190260962975698086928028977
35350061026323651913678059662436956750262280796898760568909553109913085614931
17358655518849811348977931492901697686789707664119406234554478857729840612728
45521437858746822189804434476385734388426123283329763831166365278427203525774
46330438922271136438251114020716481995174153763979626330621850596419994112592
87903736382714727871958988934812278974081264250348090322397495928698642234546
61328439994039073583026878158593023499436906078393708738818188162557767036589
57307971634668298293900963169873607225299662498839716436526166732580352360377
70337163142759843185868437864559982219949796979717015062352232344510528022042
42257826245799381897504195969580387903520894141969738681926753954872197147719
96929963271133459687072001077293048787257485321981994934718538266823393513764
95391338164081390600806728592476119592246844173337540485167177674162480442614
11771913582780994745916344839738580372517651724306625672337333894097747749699
69991247712743708922656260754521901217563615922895955152423881641986542774386
86999946535939286026523445904965654517207454772685185984888298113715301925353
98152015672953907794584778803903093961012025996566957580375807220831598864660
88722068463168337956550615211214828258395744577165820513217179830783618621689
7420865868672292434151775164
Random.org
92981853438873251392103622804762327579065952763872556655454478779613905638458
47120573526617297780522622502987448418848515475275229914395818836639257923448
54458658296265114860990111931205545860768320916981222414152050753041713957513
84791158849302632213511941917908778485517397966428305149486316865411949310017
14725686459694543528198308779477881119312671597247041856717239575526128562521
42306959907175739480179486136160408318631376984174681036322737259827183059124
47111387060569725795658981643207994506310186878232793582550795769459985585574
34998593851488378939688797716048745232092157297192030259821255163696442031588
11952930332261925975969286424513279799534819091189426792029932879427716407688
08515453612721849343715817547474165568628041762793574240790552573181869648264
72708985259874219470228965169587662745409314551765081628862190340069510794886
37234517968262388971882869788596306972807590269922835316361503225870292727244
13496839512618368855133286500710147258243224691981317019457454067516215669845
56311158608114272913796667753206376640212239238415974249552091332678813553177
55019483316289571535489112638772573867889929869831544693367252167464029332437
11795604617424003507246218275367545112520957823439506242689836895158008827413
96515195259344168565879832611369159415333413747175956792622346604202641118167
45354664706762693206833951119029765377054244344466954072886221525535221999167
65945222629146187354975759954522953169249954925375621827686171044595543961635
86650649900464223818915157801192224034945265333813933170181667124469711788691
43843536221766390890895774913894684899365245118283867129847695813437329193896
57297427598594728934881731216399793681270288537762890958353051435670911017538
53667468285157094601034055476938064338427017463218275858371130964975730615587
89727031502984370672900824875550890838871672188425826750304624565959293166329
99973675096292769317879732327546993275606609462578753471428180891842052166302
50156550457670661249987611131311969965207275887796113285013302227126957580948
81154556874174589661162371655172188829190583834447038837695280788960605260723
54940470450249634851679275779804774151483120253329457143344051395423376118258
86631828964902271121587814528835595296114083815447885325079702309574684060537
44543304564155027215871717964667072481372697192412171644807199282374555063461
94522523055361043799568058888999936325514590576819914552538019065875124395599
39858898488604165059801962372876087684179176581175111080677774223394288256696
33972117739684918931726215820308647981389945692658963838178793903306525237274
80311798888845182333419582246539722019644355745431913329638510642275443693642
42962967248101838832094329791750864459846717499838770693930589478911150922055
67457892552987888142727523071591397124930585310551941000480372720921303591957
20469674822638605367863172165526446232311016087391365772731281681974335241870
545145561852556800925502315329746237262136434
Appendix B. ENT Output
C:\temp\random>ent -b local.txt
Entropy = 0.990924 bits per bit.
Optimum compression would reduce the size
of this 23168 bit file by 0 percent.
Chi square distribution for 23168 samples is 290.88, and randomly
would exceed this value 0.01 percent of the times.
Arithmetic mean value of data bits is 0.4440 (0.5 = random).
Monte Carlo value for Pi is 4.000000000 (error 27.32 percent).
Serial correlation coefficient is 0.016484 (totally uncorrelated = 0.0).
C:\temp\random>ent -b uva.txt
Entropy = 0.991358 bits per bit.
Optimum compression would reduce the size
of this 23192 bit file by 0 percent.
Chi square distribution for 23192 samples is 277.31, and randomly
would exceed this value 0.01 percent of the times.
Arithmetic mean value of data bits is 0.4453 (0.5 = random).
Monte Carlo value for Pi is 4.000000000 (error 27.32 percent).
Serial correlation coefficient is 0.014082 (totally uncorrelated = 0.0).
C:\temp\random>ent -b ucla.txt
Entropy = 0.993697 bits per bit.
Optimum compression would reduce the size
of this 22504 bit file by 0 percent.
Chi square distribution for 22504 samples is 196.34, and randomly
would exceed this value 0.01 percent of the times.
Arithmetic mean value of data bits is 0.4533 (0.5 = random).
Monte Carlo value for Pi is 4.000000000 (error 27.32 percent).
Serial correlation coefficient is 0.007695 (totally uncorrelated = 0.0).
C:\temp\random>ent -b rand.txt
Entropy = 0.990958 bits per bit.
Optimum compression would reduce the size
of this 23032 bit file by 0 percent.
Chi square distribution for 23032 samples is 288.11, and randomly
would exceed this value 0.01 percent of the times.
Arithmetic mean value of data bits is 0.4441 (0.5 = random).
Monte Carlo value for Pi is 4.000000000 (error 27.32 percent).
Serial correlation coefficient is 0.028486 (totally uncorrelated = 0.0).
C:\temp\random>ent -b random.txt
Entropy = 0.990924 bits per bit.
Optimum compression would reduce the size
of this 23168 bit file by 0 percent.
Chi square distribution for 23168 samples is 290.88, and randomly
would exceed this value 0.01 percent of the times.
Arithmetic mean value of data bits is 0.4440 (0.5 = random).
Monte Carlo value for Pi is 4.000000000 (error 27.32 percent).
Serial correlation coefficient is 0.016484 (totally uncorrelated = 0.0).
C:\temp\random>